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| title | allDay | startTime | endTime | date | completed | type |
|---|---|---|---|---|---|---|
| Frameworks and Review | false | 12:30 | 14:30 | 2024-09-09 | null | single |
Introduction
Where do nonlinearities come from? Well, a couple of places...
- Geometric nonlinearities (pendulum)
- External fields
- Material properties So we're stuck with them. But how do we deal with noninearities?
A nonlinear equation
\dot{x} = \frac{dx}{dt} = 1-2\cos x
How do you solve this? You can't use Laplace, you can't separate... insert very long expression that Bajaj wrote. Getting an analytical solution can be a PITA to obtain. For this reason: The general case is that nonlinear equations are unsolvable. This doesn't mean we can't learn things. We can describe these systems qualitatively.
Really our options come down to:
- Solve exactly (Not likely to happen)
- Solve numerically
- Analyze qualitatively (~geometrically)
- Solve an approximation to the problem We mix and match these approaches.
Geometric (Qualitative) Methods
Geometric analysis answers questions like "is this stable?" "what's the response look like?"
Linear Systems
\dot{x} = Ax
This is a simple linear dynamic system.
How many equilibria does this system have?
One.
The system is at equilibrium where \frac{dx}{dt} = 0. It won't move from this point.
Is this system stable? Check the eigenvalues of A.
Nonlinear Systems
Recall: $\dot{x} = 1-2\cos x$